UNIVERSITÀ DEGLI STUDI DI TRENTO

Transcription

1 UNIVERSITÀ DEGLI STUDI DI TRENTO Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Specialistica in Informatica Within European Masters in Informatics Tesi di Laurea An Automatic Mapping from the Systems Biology Markup Language to the Bio-PEPA Process Algebra Relatori Prof. Corrado Priami Federica Ciocchetta Laureanda Kanimozhi Ellavarason Anno Accademico 2007/2008

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3 Abstract This dissertation concerns the automatic mapping of the Systems Biology Markup Language (SBML) into the Bio-PEPA process algebra. SBML is considered the standard for the representation, collection and exchange of biological models. Bio-PEPA is a process algebra recently defined for the modelling and analysis of biochemical systems. We develop a theoretical mapping from SBML models to Bio-PEPA systems. This is accompanied by a Java based software tool, called SBML2BioPEPA, for implementing the translation in practice. The software tool is able to successfully convert SBML models into the equivalent Bio-PEPA models. These models can then be analysed and simulated in the various tools available for Bio-PEPA models such as the Bio-PEPA Eclipse Plugin. We consider two case studies of biochemical networks to test and validate our mapping and software tool. The results we obtain are in agreement with the published results for the biological networks and with the results obtained by importing the SBML models in the COPASI software tool. Our work allows translation of SBML models to Bio-PEPA with ease. This lets researchers take advantage of the various analysis methods and software tools available for Bio-PEPA models. i

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5 Acknowledgements First and foremost, I would like to praise God for giving me the grace and blessing to get enrolled into EuMI course and giving me the strength to complete it. I have no words to thank my loving family back in India who supported me with my decision to go abroad and study. They always made me feel so close to them with their love. I would like to thank Prof. Jane Hillston for letting me work on this interesting topic. My genuine thanks to Federica, who guided me in this thesis work. I would also like to thank Prof. Corrado Priami for supervising my thesis from Trento side. My sincere thanks to all the EuMI staff in University of Edinburgh and University of Trento for I learned so much at these renowned universities. A huge thanks to all my friends for being part of these wonderful two years. And lastly, I dedicate this thesis to my soulmate...amby. Your immense Love has been has been my driving force and I know it will stay on for a lifetime! Heartfelt thanks to you, as I was able to achieve all this because of your infinite support! iii

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7 Contents 1 Introduction Introduction Motivations Objectives Structure of Dissertation Background Systems Biology Markup Language Standards in Systems Biology SBML model definition Example SBML model Process Algebras Bio-PEPA Bio-PEPA Syntax Operational Semantics Example model using Bio-PEPA Software tools Related Work Mapping from SBML to Bio-PEPA Elements of Bio-PEPA model The Mapping v

8 3.2.1 Compartments Species Parameters Functional Rates Species Components Model Component Limitations of the Mapping Summary of Mapping Software Tool Development Tools Java Classes Design Tool Deployment Case Studies Repressilator - A Synthetic Oscillatory Network Description of the model Mapping Description Simulation of Bio-PEPA Repressilator model Mitogen Activated Proetin Kinases Model Description of the model Mapping Description Simulation of Bio-PEPA MAPK model Conclusions Concluding remarks Further Work A Bio-PEPA model for the Repressilator pathway 68 B Bio-PEPA model for the MAPK pathway 71 vi

9 C Bio-PEPA model syntax for Eclipse Plugin 75 vii

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11 List of Figures 4.1 Main classes used in the SBML2BioPEPA software tool Schematic diagram of the Represillator Stochastic simulation results of oscillations in Repressilator network using Bio-PEPA plugin Stochastic simulation results of oscillations in Repressilator network using COPASI tool The Mitogen Activated Protein Kinase cascade Stochastic simulation results of MAPK using Bio-PEPA Plugin Stochastic simulation results of MAPK using COPASI tool.. 64 ix

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13 List of Tables 2.1 Operational semantics for Bio-PEPA Summary of mapping from SBML to Bio-PEPA List of Species in Repressilator model List of Parameters in Repressilator model List of Reactions in Repressilator model List of Species in MAPK model List of Parameters in MAPK model List of Reactions in MAPK model xi

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15 Chapter 1 Introduction This chapter presents an introduction to the main topics of the thesis. The thesis is devoted to the design and development of a software tool for automatic translation of models from Systems Biology Markup Language (SBML) to the process algebra Bio-PEPA. After an introduction to the context of application of our work, we describe the motivation of our research and the objectives we aim to achieve. Finally, the structure of the thesis is reported. 1.1 Introduction Molecular biology techniques have unveiled enormous amounts of information on genes and proteins. The information obtained is at the molecular level. At the system level, genes and proteins interact in a very complex fashion. The challenge is now to understand the functions and the behaviour of the system as a whole. This is the aim of Systems Biology, a relatively new field of study. As described in [1], Systems biology deals with the systems level understanding of the interactions in the biological systems. Computational modelling and simulation of biological pathways are important research aspects in systems biology. They can help us understand the internal nature and the dynamics of the process and arrive at well founded 1

16 predictions about the future developments [2]. There are numerous techniques for the modelling and analysis of biological systems. Some of the important techniques that have been used for modelling are ordinary differential equations [3], Petri nets [4], process algebras [5], Boolean networks [6] and graph based approaches [7]. Recently, there has been increasing interest in the application of process algebras in the field of systems biology [8], [9], [10], [11], [12], [13], [14], [15], [16]. Process algebras are mathematical formalisms originally defined for the modelling and analysis of concurrent systems [5], [17]. The entities such as ion or molecules in the biosystems interact in a concurrent fashion. Hence, the various species can be abstracted in a process algebra context by processes interacting with each other by means of actions [8]. Process algebra formalisms are useful in systems biology as they offer a formal representation of the model, a compositional approach and support different kinds of analysis. Examples of process algebras applied to biology are π-calculus [5], stochastic-π calculus [19], Beta-binders [9], Performance Evaluation Process Algebra (PEPA) [20] and Bio-PEPA [16]. Among the different process algebras, in this dissertation we consider Bio-PEPA. Bio-PEPA extends the process algebra PEPA with some constructs to render features of biochemical networks. Another important research area in systems biology is the definition of a standard for the representation of biochemical networks. Among the numerous proposals, SBML [22] is emerging as the standard for the representation and exchange of biological models. It is supported by many software tools and has gained widespread acceptance [10]. 1.2 Motivations The aim of this dissertation is the definition of an automatic conversion of models from SBML to Bio-PEPA. The main motivations of this work are 2

17 mentioned below: Among the different exchange languages available, SBML is emerging as the standard for the representation of biological models. It provides a way to exchange models between different softwares without the need of re-encoding the model. It is well accepted and is in widespread use. It is known to be supported by over 90 software packages. Further, a lot of models have been collected in SBML format, such as the ones in the Biomodels database [23]. Due to these reasons, we have chosen SBML in our research. Process algebras are formalisms used to represent concurrent systems. They have properties that make them particularly useful in the context of systems biology such as compositionality, formal representation, simple abstraction and analysis. Among the different process algebras we consider Bio-PEPA for our dissertation. There are various motivations for this choice. Bio-PEPA is specifically designed for representing biochemical systems. In particular, it is possible to represent stoichiometry and general kinetic-laws (different from mass-action). It supports numerous kinds of analysis and therefore allows us to investigate the properties of the system from different compatible views. We can perform stochastic and deterministic simulation, check the formal properties of the system by using model checking or consider other analysis based on continous time Markov chain (CTMC). Bio-PEPA is a formal language and it is not straight-forward for nonexperts to model biosystems using its constructs. Developing a tool for automatic conversion from SBML to Bio-PEPA will enable users to generate Bio-PEPA models with ease. In this way the complexity of the formalism can be hidden from the end user. 3

18 The correctness of the tool can be tested by translating some wellknown SBML models into Bio-PEPA using our tool and performing simulations on the translated models. The results obtained can then be compared with the results published in the literature and the results obtained by importing the same SBML models into other analysis tools supporting SBML. 1.3 Objectives The main goals of this dissertation are: 1. To present a formal translation from SBML to Bio-PEPA. We discuss the theoretical mapping and also mention any assumptions we make. 2. To develop a Java based software tool for translation of the SBML models to Bio-PEPA. The tool derives a Bio-PEPA specification corresponding to the SBML model. The syntax is the one that is used as input for the Bio-PEPA plugin tool for analysis. 3. To generate Bio-PEPA models of biological systems found in literature using the above tool. These models will then be simulated using the Bio-PEPA plug-in and the obtained results will be compared with the published ones. Furthermore, the model can be used to investigate further properties and possible behaviours of the associated biological system. 1.4 Structure of Dissertation This chapter described the introduction, motivations and the objectives of our work. The rest of the dissertation is structured as follows. Chapter 2 reviews the background of SBML with a description of its basic elements. We also describe Bio-PEPA and its usage in modelling biological systems. In 4

19 Chapter 3, we put forward the theoretical mapping and assumptions for the translation of models from SBML to Bio-PEPA. Chapter 4 talks about the class design and data structures of the software tool which implements the mapping. In Chapter 5, the results obtained using the software tool in two case studies are elucidated. The results obtained are analysed and compared with the literature. Finally, in Chapter 6, major conclusions of our work are discussed and future work directions are proposed. 5

20 Chapter 2 Background In this chapter we describe the background of our work. We begin by describing the necessity of standards for the representation of biological systems followed by a detailed description of SBML and its components. A small example of an SBML model is given in order to illustrate the language. We then introduce process algebras and explain their advantages. The Bio-PEPA syntax used for modelling biological systems is given along with a description of semantics, and an example of a Bio-PEPA model is presented. We then briefly mention the software tools we have used in our work. Finally, some related works concerning the translation of SBML into process algebras are reported. 2.1 Systems Biology Markup Language The Systems Biology Markup Language (SBML) [22] is a machine-readable format for describing qualitative and quantitative models of biochemical networks. SBML is based on the extensible Markup Language (XML) [24], a popular text-based language for expressing structured data in a generic fashion. It enables system biologists to share, evaluate and develop models cooperatively between different formalisms. The requirement of standard models 6

21 like SBML is described in the next subsection Standards in Systems Biology Computational modelling techniques use various software tools for understanding the biological systems. There are many different formats of models available. Systems biologists find it difficult to work with different formats. The need of a standard representation of models is important in systems biology because of the following reasons: 1. Users have to use multiple tools in their research. Different software tools have different strengths and capabilities. However, these tools cannot exchange their models because of having their own particular representation formats. 2. Models that are published may use different modelling environments and model representation languages. To use the model, users have to manually translate the model into the particular formats they use. 3. Models could be stranded if simulation is not supported by the model representation language. This will result in the loss of useful models. A standard format representation is desirable in this case. Due to these reasons, the need for the standard representation of models is an important issue. SBML has been developed in an effort to address the above problems. SBML is evolving as a de facto standard for the common representation of biological models. In this way, the diversity of approaches used by different software tools is acknowledged and a common format is followed. Among the different exchange languages, SBML is the most known and used. The popularity of SBML is growing in systems biology community as many software tools are supporting it and models are being collected in model repositories such as 7

22 the BioModels database [23]. The SBML community continues to develop and make software environments including programming libraries, conversion utilities, interface packages for commonly used software environments and easy to access online tools. SBML is developed in levels; two levels have been defined so far. SBML Level 1 is the basic version and has lesser representational power than SBML Level 2 where the model definitions and mathematical expressions have been enriched. Software tools that cannot or do not need to represent higher levels can use Level 1. In this thesis, we have considered the latest available version of SBML which is Level 2, Version SBML model definition A biochemical network consists of compartments, species, reactions (including reactants, products, reaction rate expressions and parameters). SBML describes these components and additional information by using tags and attributes [25]. A schema of an SBML model is presented below and each component is described in detail. <model id="my_model" > <listoffunctiondefinitions>... </listoffunctiondefinitions> <listofunitdefinitions>... </listofunitdefinitions> <listofcompartments>... </listofcompartments> <listofspecies>... 8

23 </listofspecies> <listofparameters>... </listofparameters> <listofrules>... </listofrules> <listofreactions>... <listofreactants>... </listofreactants> <listofproducts>... </listofproducts> <listofmodifiers>... </listofmodifiers> <listoflocalparameters>... </listoflocalparameters> </listofreactions> <listofevents>... </listofevents> </model> An SBML model definition consists of lists of SBML components. The instances of the classes ListOfComponents must be located inside the tags <listofcomponents>... </listofcomponents> 9

24 where components could be FunctionDefinitions, UnitDefinitions, Compartments, Species, Parameters, Rules, Constraints, Reactions and Events. Although all lists are optional, if a given component is present in the model then the corresponding list cannot be empty. The definition of one component can have dependencies on other component definitions. For example, the definition of reactions is built on other SBML elements, such as species and compartments. Mathematical expressions such as kinetic-laws, stoichiometry, events, functions, rules occur in the model definition. The representation and semantic interpretation is done using MathML [27] which is the international standard for encoding mathematical expressions using XML. We describe below the main SBML components and their principal tags and attributes: Function definition - The mathematical functions that can be used in the other part of the model are defined. Function defintions are done using MathML expressions. Function definitions contain the following important attributes: 1. Function definition id 2. MathML expression Unit definition - Units are mathematical entities that are specified explicitly. The definitions of sizes of the compartments, initial amounts of species, constant units and variable parameter values have units associated with them. SBML defines units using built in functions. The main attributes of unit definition are: 1. Unit id 2. Unit name 3. List of units 10

25 Compartment - Species generally exist in a finite bound space known as compartment. Every species that is defined in the model must be located in a compartment. If there are any species defined in the model, it is mandatory that at least one compartment type is defined. The attributes of compartment are: 1. Compartment id 2. Compartment name 3. Compartment type 4. Compartment size Species - Simple entities (such as molecules or ions) that take part in a chemical reaction are known as species. They are the reacting entities of the specific species type. The initial quantity of the species are set in the SBML file. If the quantity is missing, it is either unavailable or set by an assignment rule. We list the important attributes that we consider for the translation: 1. Species id 2. Species name 3. Species type 4. Compartment name 5. Initial concentration Reaction - Biochemical species interact with each other in reactions. The speed of a reaction is expressed in terms of a rate-law involving species and parameters. For each reaction the SBML model includes the details about the species involved, their stoichiometric coefficients, the parameters and the rate-law. The species can be reactants, products or modifiers (i.e. species which take part in a reaction but are 11

26 neither consumed nor produced as part of the reaction). The SBML attributes and sub-tags for reaction are: 1. Reaction id 2. Reaction name 3. List of reactants and their stoichiometries 4. List of products and their stoichiometries 5. List of modifiers 6. Kinetic-law expressed by MathML expression Parameter - A parameter is a variable used in mathematical expressions. Parameters in SBML can be of two kinds: global and local. Parameters that are defined as global are available to the whole model. The parameters defined as local are independent of the global parameters and are defined for particular reactions. The attributes are: 1. Parameter id 2. Parameter name 3. Parameter value 4. Parameter units Rule - Apart from the local and global parameters that are defined in the model, rules provide a way of defining the value of variables in the model, and the dynamic behaviour of those variables. Rules express the relationships that cannot be expressed using reactions alone. The list of attributes for rules are: 1. Variable name 2. MathML expression used to express the formula. 12

27 Event - Events are constructs that represent changes in the system due to some trigger conditions. During the evolution of the system, events are evaluated using the mathematical formulas at specified time intervals. It enables the modelling of various biological phenomena. The attributes are: 1. Event id 2. Event name 3. Trigger 4. Delay 5. List of event assignments using MathML expressions Example SBML model Consider the system composed of species A, B, C and E involved in the following reaction. A + B E, k1[a][b] 2C Here, A and B are the reactants, C is the product and E is the modifier. [A] and [B] represent the concentrations of A and B respectively. The stoichiometries of A, B and C are 1, 1 and 2 respectively. The sample SBML file for this simple reaction is: <?xml version="1.0" encoding="utf-8"?> <sbml xmlns=" level="2" version="3"> <model id="sbmlsample" name="sbmlsample"> <listofunitdefinitions> <unitdefinition id="time" name="minutes"> 13

28 <listofunits> <unit kind="second" multiplier="60" /> </listofunits> </unitdefinition> </listofunitdefinitions> <listofcompartments> <compartment id="cell" name="cell" size="1" units="volume"/> </listofcompartments> <listofspecies> <species id="a" name="a" compartment="cell" initialamount="1" /> <species id="b" name="b" compartment="cell" initialamount="1" /> <species id="c" name="c" compartment="cell" initialamount="0" /> <species id="e" name="e" compartment="cell" initialamount="1" /> </listofspecies> <listofreactions> <Reaction id = "R1"> <listofreactants> <speciesreference species = "A"/> <speciesreference species = "B"/> </listofreactants> <listofmodifiers> <speciesreference species = "E"/> </listofmodifiers> <listofproducts> <speciesreference species = "C" stoichiometry = "2"/> </listofproducts> <Kineticlaw> 14

29 <math xmlns = " <apply> <times/> <ci>k1</ci> <ci>a</ci> <ci>b</ci> </apply> </math> <listofparameters> <parameter id="k1" value ="100" units ="litre_per_mole_per_second"/> </listofparameters> </kineticlaw> </reaction> </listofreactions> </sbml> The model has the identifier SBMLsample. It contains one compartment with identifier cell and four species with identifiers A, B, C and E and one reaction R1. These elements corresponds to the species and the reaction of the biochemical system. The elements listed in ListOfReactants and ListOfProducts in each reaction refer to the names of elements listed in ListOfSpecies. The references between the species have been illustrated by Species reference elements. The declaration of local parameters such as k1 has been done in the model. 2.2 Process Algebras Process algebras are mathematical formalisms used to describe and analyse concurrent systems [5], [17]. A concurrent system is composed of multiple processes running together. These multiple processes can either run independently or synchronise their activities with other processes. Biological 15

30 systems can be viewed as concurrent systems where molecules are interacting processes and reactions can be represented by interactions between these processes [8]. We describe some features of process algebras that are useful in the context of systems biology: 1. Compositionality: The whole system can be defined starting from the definition of its sub components. 2. Formal meaning: Process algebras are mathematical formalisms, they are based on precise syntax and their semantics are clearly defined. Hence, models can be unambiguosly defined. 3. Abstraction: Process algebras can be used for modelling at various levels of abstraction. It is easy for modellers to model at the level that they require. 4. Analysis: Process algebras support various kinds of analysis. For example, it is possible to study the causal relationship of the components or model checking and validating the model and correcting any inaccuracies [18]. 2.3 Bio-PEPA In this thesis we consider Bio-PEPA [16], [12], a process algebra recently defined for the modelling and analysis of biochemical networks. It refers to PEPA [20] and extends it with some constructs to represent some specific biological features. PEPA is a process algebra originally defined for performance modelling of computer systems and has been recently applied for modelling of signalling pathways, see for instance [11]. In this work two different modelling styles are proposed: 16

31 Reagent centric view: A discrete number of integer levels is associated with each chemical species; with each level representing a concentration value (between the minimum and maximum possible concentration values) for that species. Each of these levels has an associated PEPA process and the system model is formed by forcing these processes (species) to cooperate over the reactions they are involved in. So for a reaction to happen, all the processes (species) must be enabled. Pathway centric view: This is similar to the above, but it represents a more abstract view where the processes represent sub-pathways instead of individual species. PEPA is appropriate for modelling biochemical pathways, but not all features of biochemical networks can be represented using this process algebra. Two main problems are: 1. Stoichiometric coefficients different from one cannot be represented. Stoichiometry represents the quantitative relationships of elements involved in the reaction. 2. Kinetic-laws different from the simple mass-action (hereafter called the general kinetic-laws) cannot be specified. These two features are found in many biochemical networks. Bio-PEPA has been designed in order to represent them. The information about stoichiometry is added to the syntax and the kinetic-laws are expressed by using functional rates. Note that Bio-PEPA refers to the reagent centric view of PEPA. In the next subsection, we consider the syntax of Bio-PEPA in detail. 17

32 2.3.1 Bio-PEPA Syntax The syntax of Bio-PEPA [16] consists of sequential component and model component. These are defined as: S ::= (α, k) op S S + S C P ::= P L P S[l] The component S is the sequential component and represents a biochemical species, P is the model component and defines the interactions among sequential components [16]. The various operators are explained below: Prefix: The prefix term is represented as S ::= (α, k) op S α represents the action type (i.e. a unique name associated with each reaction in the system) and k represents the stoichiometry of the species in the reaction. The combinator op represents the components role. represents reactant, means product, means an activator, an inhibitor and a generic modifier respectively. Choice: The choice operator is represented by S + P. It means a component which can behave either as S or as P. Both the processes start their activities concurrently. However, the component that completes the activity first is chosen and the other is discarded. Constant: C represents a constant component defined by the equation S = C. Constants derive their meaning from the defining equation. For example, the behaviour of S can be assigned to C. It allows us to assign names to components. Cooperation: The term P Q is used to depict when P and Q interact L 18

33 with each other to carry out an activity together. The common activities are listed in set L of activities. If an activity is listed in this set, then the components are forced to synchronise on that activity. The activities outside set L proceed independently. Note that Bio-PEPA supports multiway synchronisation, wherein more than 2 components can cooperate on a given activity. Levels of concentration: The term S[l] denotes the number of levels of each species, where l is the discrete levels of concentration. We now give the definition of a Bio-PEPA system as described in [16]. A Bio-PEPA system P is a 6-tuple V, N, K, F R, Comp, P, where: V is the set of compartments in the model. Compartments are static (i.e. with a fixed structure and size) and the set of compartments is assumed to contain at least one compartment. N is a set of quantities associated with each species. This set consists of the following elements: a component name C, step size H, maximum number of levels N, name of the enclosing compartment, initial concentration M 0, maximum concentration M, and the unit for the species concentration. The last 4 elements are optional in the definition of a species. K is the set of parameter definitions. F R is the set of functional rate definitions. This describes the set of kinetic-laws of the reactions in the model. Comp is the set of definitions of sequential components. P is the model component describing the overall system. 19

34 prefixreac ((α, κ) S)(l) (α,[s: (l,κ)]) c S(l κ) κ l N prefixprod ((α, κ) S)(l) (α,[s: (l,κ)]) c S(l + κ) 0 l (N κ) prefixmod ((α, κ) op S)(l) (α,[s:op(l,κ)]) c S(l) with op =,, and 0 < l N if op =, 0 l N otherwise choice1 S 1 (l) (α,w) c S 1(l ) (S 1 + S 2 )(l) (α,w) c S 1(l ) choice2 S 2 (l) (α,w) c S 2(l ) (S 1 + S 2 )(l) (α,w) c S 2(l ) constant S(l) (α,s:[op(l,κ)]) c S (l ) C(l) (α,c:[op(l,κ)]) c S (l ) with C def = S coop1 coop2 coop3 Final P 1 (α,w) c P 1 P 1 P (α,w) 2 L c P 1 P 2 L P 2 (α,w) c P 2 P 1 P (α,w) 2 L c P 1 P L 2 P 1 (α,w 1 ) c P 1 P 2 (α,w 2 ) c P 2 P 1 P (α,w 1 ::w 2 ) 2 L c P 1 P L 2 P (α j,w) c P P (α j,f α(w,n )) P with α / L with α / L with α L Table 2.1: Operational Semantics for Bio-PEPA Operational Semantics Structural operational semantics [21] defines the the behaviour of a process algebra system in terms of a set of inference rules. These rules describe the 20

35 valid transitions of each element in the syntax in terms of the transitions of its components. The operational semantics for Bio-PEPA systems are given in Table 2.1 [16]. The first three prefixes define the behaviour of prefix terms reactant, product and modifier. In case of reactant the level decreases, in case of product the level increases and in modifiers the level remains the same. The rules choice 1 and choice 2 define that any one of the two process can start the activity. The constant rule defines the behaviour of constant term. The coop1, coop2 and coop3 rules depict the case of co-operation. The first two rules predict the behaviour when action is enabled and does not belong to the cooperation set. The rule coopfinal describes the case in which two components synchronize and information from labels of both the components are available. The rule Final is included to represent the rate associated with the transition. The second component f α (w, N) in the label of the conclusion means that the function f α is evaluated over the list of quantitative information w and the set N of maximum concentration/number of levels. The reader is referred to [16] for additional information on the structural operational semantics of Bio-PEPA Example model using Bio-PEPA We consider the example that we used to illustrate SBML models in Section To represent this in Bio-PEPA, we define a sequential component for each species in the model as follows: We associate a unique reaction name with each reaction in the system. We then define the sequential component for each species. If a species participates in a reaction, it performs an activity which is the reaction name. The stoichiometry and role of the species in that reaction are represented with the syntax described in the previous section. If a species is participating in multiple reactions, then this represented by different activities (reaction names) combined with the choice operator. 21

36 Finally, we define the system component to include all the species such that they are forced to synchronise on the reaction names they take part in. The system component also includes information about the initial level of each sequential component. The Bio-PEPA system for the reaction system shown above is: Species Definitions: A = (R1, 1) A B = (R1, 1) B C = (R1, 2) C E = (R1, 1) E Parameter: k1 = 100 Kinetic-Law: f R1 = fma(k1) Model Component: A[1] {R1} B[1] {R1} C[0] {R1} E[1] In this Bio-PEPA model, the sequential component of each species is defined with the reaction name, stoichiometry and role of the species in the reaction. The constant parameter is defined as k1 = 100. In the functional rate definition, fma(k1) stands for the mass-action kinetic-law with constant parameter k1. The level of each species is denoted by specifying S[l] in the model component. For instance, we have used A[1], B[1], C[0] and E[1] respectively. 22

37 2.4 Software tools In this section we describe the main software tools we have used in our work. Eclipse IDE: We use Eclipse IDE in our work for writing the source code of the software tool used to convert models from SBML to Bio- PEPA. Eclipse is a powerful software platform comprised of extensible application frameworks, tools and runtimes for software development and management, primarily written in Java and an open-source Integrated Development Environment (IDE) [28]. It has a lot of features supporting effective Java development such as incremental Java compilation, advanced editing and highlighting of code and syntax errors and the ability to install various plugins for additional features. Bio-PEPA Plug-in: The modelling of the Bio-PEPA models created by our software tool was done in the Bio-PEPA Eclipse plug-in. It accepts Bio-PEPA files as input and allows us to run deterministic as well as stochastic simulations of the Bio-PEPA model. We have considered the latest available version of the tool. The tool supports deterministic and stochastic simulations and the implementation of other kinds of analysis is under development. COPASI: COmplex PAthway SImulator (COPASI) [29] is a widely-used software tool for simulation and analysis of biochemical networks. We have used COPASI to import and perform simulations of the SBML models of the pathways we have considered in this thesis. This allows us to compare the simulation results obtained from the tool and therefore to validate our translation. 23

38 2.5 Related Work There have been some previous works concerned with mapping SBML to process algebras such as: SBML to π-calculus (Bio-SPI tool) [32] SBML to stochastic π-calculus (SBML2Pi tool) [33] SBML to Beta-binders [18] Each of the above mappings varies in the level of information that is present in the translated model. They also differ in the assumptions and the methods that they follow. All the above mappings have some limitations. First, only reactions with order up to two can be specified. Indeed, in both pi-calculus and beta-binders, reactions are abstracted by communications between processes representing species, and communications are pairwise. Secondly, as a consequence of the first point, stoichiometric coefficients cannot be represented. Third, these process algebras assume that the kinetic-law must be of kind mass-action, whereas rate-laws other than mass-action are common in biological systems. Finally, analysis by using Gillespie s algorithms [31] for stochastic simulation is generally possible but further analysis using different approaches is not. Note that that both pi-calculus and Beta-binders have been recently extended with biological transactions in order to handle multiple-reactant multipleproduct reactions [34] [35], but these extensions have not been implemented in the analysis tools yet, and in the case of the pi-calculus this extension is not considered in the mapping. We also point out that BlenX [10], a language used for modelling biological processes, is based on Beta-binders and support functional rates but in [18] the standard Beta-binders is used. Bio-PEPA allows us to represent all these aspects of biochemical systems. 24

39 Indeed it supports multiway synchronisation so it is possible to map any reaction, including the ones with order greater than two. In addition to this, Bio-PEPA supports the definition of stoichiometry and the use of functional rates to express general rate-laws. Bio-PEPA models also support several kinds of analysis such as ODE-based, stochastic simulations, continuous time Markov chain based analysis and model checking. 25

40 Chapter 3 Mapping from SBML to Bio-PEPA The final aim of our work is to map biochemical systems described in SBML into Bio-PEPA. This allows us to study the systems by using the various kinds of analysis supported by this process algebra. In this chapter, we describe the basic elements that are required to build a well-formed Bio-PEPA model. Then, we explain how these building blocks of Bio-PEPA models can be obtained from the information in the SBML file. We also discuss the assumptions we have made and the additional information required for the successful conversion to Bio-PEPA, as well as the elements which cannot be mapped into Bio-PEPA. The chapter ends with a summary of the main features of the mapping from SBML to Bio-PEPA. 3.1 Elements of Bio-PEPA model A well-formed Bio-PEPA model is defined by a set of compartments, a set of species, a set of parameters, a set of functional rates, species components and a model component. We briefly describe below the information required to build each of these elements. The description of each component is as 26

41 reported in [16]. 1. Compartments: A compartment is described by V: v unit where V is the compartment name, v is a positive real number expressing the compartment size and unit (optional) denotes the unit associated with the compartment size. 2. Species: Species are defined by a name C, number of levels N, step size H, initial concentration M 0, maximum concentration M, the enclosing compartment name V and the (optional) unit for the species concentration. 3. Parameters: Each parameter has an associated name and value, and an optional unit. 4. Functional rates: There is one functional rate associated with each reaction, which specifies a mathematical formula denoting the rate of that reaction. 5. Species components: A Bio-PEPA sequential component abstracts a species in the biochemical system. Each component specifies the reactions which the species participates in, and the stoichiometry and the role of the species in the reactions. 6. Model component: This is formed by forcing the species components to cooperate over the reactions they are involved in. The model component also includes information about the initial level of each species. 3.2 The Mapping In this section, we describe how to define each of the Bio-PEPA model elements listed in Section 3.1 from the information in the SBML file. Furthermore we discuss the assumptions and limitations of our approach. 27

42 3.2.1 Compartments 1. SBML files contain a section called ListOfCompartments. Each compartment in this list is directly mapped to a compartment in Bio-PEPA. SBML compartments have a unique ID attribute. This ID is mapped to the compartment name ( V ) in Bio-PEPA. SBML compartments also have an optional attribute called size. This is mapped to the compartment size ( v ) in BioPEPA. If the size is not specified in the SBML file, we set a default size of 1 in the Bio-PEPA model. If a unit attribute is specified in the SBML model, we map this to the unit in Bio-PEPA. However, the unit is optional in Bio-PEPA as well, so we do not specify any default value if it is not present in the SBML file. 2. Strictly speaking, list of compartments is optional in SBML. If there are no compartments specified, then we add a default compartment of size 1 in Bio-PEPA. However, if the SBML file defines any species, then it also has to define at least one compartment since each species must belong to a compartment in SBML. Hence for all models in practice SBML files have at least one compartment. 3. The Bio-PEPA Eclipse plugin version we considered does not allow us to define a compartment whose id is compartment, although this is valid in SBML. If we find a SBML compartment with name compartment we simply rename it to compartment biopepa. 4. We give examples to explain the above points. Consider a SBML file which contains a ListOfCompartments as below: <ListOfCompartments> <compartment id = "A" size = "1" units = "volumeunits" /> <compartment id = "B" size = "2" /> <compartment id = "C" /> </ListOfCompartments> 28

43 These compartments are converted to Bio-PEPA as below. compartment A : 1 volumeunits compartment B : 2 compartment C : 1 Here, for compartment A the SBML model has all the required information. For compartment B, the units are not listed but these are optional in both SBML and Bio-PEPA so we do not specify any default. For compartment C, we assume a default size of 1 (the units depend on the particular model) Species 1. SBML files contain a section called ListOfSpecies. Each species in this list is mapped to a species in Bio-PEPA. The SBML definition of species may include attributes for ID, initialconcentration, compartment and unit. These attributes are mapped in Bio-PEPA to the species name C, the initial concentration M 0, the enclosing compartment name V and the unit for the species concentration respectively. 2. If the initialconcentration is not specified in the SBML file, then it must be obtained from external sources and inserted in the Bio-PEPA model. Units are optional attributes in both SBML and Bio-PEPA. If units are not specified in the SBML file, we do not specify any default value in the Bio-PEPA model. 3. The information about the number of levels N, the step size H and the maximum concentration M cannot be obtained from the SBML file. We specify a default value of 1 for the step size H and 2 for the number of levels N in the Bio-PEPA model. The maximum concentration is optional in Bio-PEPA, so we do not specify any default value for it. The 29

44 correct values for these elements need to be obtained from external references if they are required for modelling. Note that the values for these elements are not required if we only want to run ODE-based or stochastic simulations. They are needed only for CTMC-based analysis and model checking. 4. The version of the Bio-PEPA Eclipse plugin we considered does not allow us to use names such as M, H, M0 etc which are used for defining the properties of species. If we find a species with such a name, we simply append species to the name. 5. We now give a simple example to explain the above points. Consider a SBML file which has the following ListOfSpecies: <listofspecies> <species id = "S1" compartment="a" initialconcentration = "1" units = "concunits" /> <species id = "S2" compartment="b" /> </listofspecies> These species are converted to Bio-PEPA as below. species S1 : M0 = 1, V = A, H = 1, N = 2, unit = concunits species S2 : V = B, H = 1, N = Parameters 1. SBML files can contain ListOfParameters in two places: one global list and another local parameters list included within the KineticLaw element of a reaction. For each parameter in these lists we create a Bio-PEPA parameter. The ID attribute is mapped to the name and the value attribute is mapped to the value of this parameter in the Bio-PEPA model. The value attribute is optional in SBML; if not 30

45 specified, it must be obtained from external references and inserted into the Bio-PEPA model. 2. Parameters local to a reaction can share the same name across different reactions, but can have different values. However in Bio-PEPA we can have only one parameter with a given name and its value. So, for local parameters, we append the reaction ID to the parameter name to differentiate local parameters of different reactions which have the same name. 3. The units of a parameter are optional in both SBML and Bio-PEPA. So, we do not specify any default unit if it is not specified in the SBML file. 4. We explain the above with the help of an example. <!-- Global parameter list --> <listofparameters> <parameter id = "k1" value = "100" /> </listofparameters> <!-- Local Parameter list for reaction id R1 --> <listofparameters> <parameter id = "k2" value = "200" /> </listofparameters> <!-- Local Parameter list for reaction id R2 --> <listofparameters> <parameter id = "k2" value = "300" /> </listofparameters> 31

46 These parameters are converted to Bio-PEPA as below. k1 = 100 k2 R1 = 200 K2 R2 = 300 The parameter k1 is mapped directly to the corresponding Bio-PEPA parameter. Both reactions R1 and R2 have a local parameter called k2, we append the reaction name to the parameter to differentiate them Functional Rates 1. SBML files contain a ListOfReactions. For each of the reactions in this list, we create a name called Alpha N where N is an integer number denoting the number of the reaction in the ListOfReactions. We then associate a rate formula with this reaction name. This is obtained from a sub-element called KineticLaw of the reaction in the SBML file. It represents the rate of the associated reaction. 2. The KineticLaw for a reaction in SBML is optional, however there is no default rate formula we can use if it is not specified. For those reactions which do not have a KineticLaw specified, the user will have to obtain the rate formula from external references and insert this in the Bio-PEPA model. 3. A reaction can also have an attribute called reversible which specifies whether the reaction is reversible or irreversible. If the value of this attribute is set to true, then we split the reaction into two irreversible reactions denoting the forward and backward reactions respectively and proceed as above to generate the functional rate. 4. Note that the Bio-PEPA syntax allows us to use pre-defined rate formulas such a fma, fmm and fh for mass-action, Michaelis-Menten 32

47 and Hill kinetics respectively. Although the kinetic-law of a reaction in SBML might be based on one of these kinetics, it is not possible to get this information from the SBML file. Hence the Bio-PEPA model will always contain the expanded formula in terms of species and parameters, instead of just the parameter being used in one of the pre-defined functions available in Bio-PEPA. 5. Consider a simple example of an SBML file which contains the following reaction: <Reaction id = "R1"> <!-- Reactants, products, modifiers etc not listed --> <Kineticlaw> <math xmlns = " <apply> <times/> <ci>k1</ci> <ci>s1</ci> <ci>s2</ci> </apply> </math> </KineticLaw> </Reaction> The Bio-PEPA functional rate corresponding to this reaction is specified as below. kineticlawofalpha 1 : k1 S1 S2 33

48 3.2.5 Species Components 1. SBML files contain a ListOfReactions. For each reaction, we create a name called Alpha N where N is an integer number denoting the number of the reaction in the ListOfReactions (this name is the same as the one we used for the functional rates in the earlier subsection). 2. We then create a Bio-PEPA sequential component for each species in the SBML ListOfSpecies. Each reaction in the list of reactions may contain a ListOfReactants, ListOfProducts and ListOfModifiers. If a species occurs in the ListOfReactants/Products/Modifiers of a reaction, then it performs an action whose name is the one associated with that reaction ( Alpha N from the above point). The species reference in the three lists also specifies the stoichiometry of the species. Depending on which list (reactants, products or modifiers) a species occurs, its role is specified as (reactant), (product), and (generic modifier) respectively. 3. SBML does not have a way of representing activators or inhibitors. Both are grouped under the ListOfModifiers. Hence it is not possible to specify the exact role of a species as (activator) or (inhibitor) based on the information in a SBML file. 4. Again, we explain this with the help of a simple example. Consider a list of species and reactions in SBML as below. <listofspecies> <species id = "S1" compartment = "A" initialamount="1" /> <species id = "S2" compartment = "A" initialamount="0" /> <species id = "S3" compartment = "A" initialamount="1" /> </listofspecies> <listofreactions> 34

49 <Reaction id = "R1"> <listofreactants> <speciesreference species = "S1" stoichiometry = "2"/> </listofreactants> <listofproducts> <speciesreference species = "S2"/> </listofproducts> <!-- Kinetic-Law and Parameters now shown --> </Reaction> <Reaction id = "R2"> <listofreactants> <speciesreference species = "S2"/> </listofreactants> <listofmodifiers> <speciesreference species = "S3"/> </listofmodifiers> <listofproducts> <speciesreference species = "S1"/> </listofproducts> <!-- Kinetic-Law and Parameters now shown --> </Reaction> </listofreactions> The species definitions corresponding to this SBML file is as follows: S1 = (R1, 2) S1 + (R2, 1) S1 S2 = (R1, 1) S2 + (R2, 1) S2 S3 = (R2, 1) S3 Species S1 participates in R1 and R2 as reactant and product respectively. S2 participates in R1 and R2 as product and reactant respectively. S3 participates in R2 as a modifier. 35

50 3.2.6 Model Component 1. The model component includes every species in the ListOfSpecies in the SBML file. Assume that the species in this list are numbered from 1 to N. We include species i in the model component. This is followed by a list of reactions which species i is forced to synchronise on with species i + 1 to N. In other words, if a reaction involves species i and any other species i + 1 to N then the reaction is included in the synchronisation list listed after species i. 2. Each species should also have an appropriate initial level in the model component. However, this information cannot be obtained from the SBML file. We specify a default initial level of 1 for each species in the model component. The initial level does not matter when we are dealing with ODE-based or stochastic simulation of the Bio-PEPA model. If a user is interested in performing CTMC based analysis on the model, he can set the initial level to be the largest integer value which is less than or equal to M 0 /H, i.e. the initial concentration divided by the step size [16]. 3. We explain how to build the model component, again considering the SBML example from Sub-section Considering the species in order, we first include species S1 with initial level 1. S1 participates in reaction R1 with S2, and in reaction R2 with both S2 and S3. Hence both R1 and R2 are included in the synchronisation set listed after S1. We then list species S2 at initial level 1. S2 participates in reaction R2 with S3. Hence we list R2 in the synchronisation set listed after S2. We then included the only remaining species S3 in the model definition, which is then complete. The final model component definition is as follows: S1[1] S2[1] S3[1] {R1,R2} {R2} 36

51 3.3 Limitations of the Mapping We list below some of the limitations of our mapping from SBML to Bio- PEPA as described in Section The mapping does not consider SBML events. This is because the Bio-PEPA syntax we have considered does not support the inclusion of events. However, there has recently been work on incorporating SBMLlike events into Bio-PEPA [36], and in the future we would be able to include ListOfEvents in our mapping as well. 2. We have ignored the ListOfRules and ListOfInitialAssignments in SBML, since parameter values and species concentrations in Bio-PEPA are assumed to be initially constant, and not evaluated by functions either at start or during the course of a model simulation. 3. We are not able to translate certain SBML attributes meaningfully into Bio-PEPA. For example, we cannot map attributes such as constant = false for SBML compartments, since compartments in Bio-PEPA are considered to be static. However, an extension of Bio-PEPA to handle compartments of variable size has recently been defined [37], and this can be integrated into the mapping in the future. 37

52 3.4 Summary of Mapping We summarise here the mapping from SBML elements to Bio-PEPA elements. SBML Element Corresponding Bio-PEPA component List of Compartments Bio-PEPA compartments List of Species Species definitions (Name, initial concentration and compartment). Step size and level default to 1. Also used in species sequential component definitions. List of Parameters Bio-PEPA parameter list. Local parameters renamed to include reaction name. List of Reactions Species component definitions and model component definition. Kinetic-Laws Rules, Events, InitialAssignments etc. Bio-PEPA Functional rates Not mapped to Bio-PEPA Table 3.1: Summary of mapping from SBML to Bio-PEPA 38

53 Chapter 4 Software Tool This chapter is devoted to the description of the tool called SBML2BioPEPA which implements the mapping from SBML to Bio-PEPA as discussed in Chapter 3. We begin by describing the reasons for our choice of development tools. We then describe the main software classes created for the development of the tool. 4.1 Development Tools Our software tool SBML2BioPEPA has been implemented in the JAVA programming language [38]. The Eclipse IDE has been used for development. The reasons for our choices are reported below: 1. The most important reason for this choice of language was to enable closer integration with other Bio-PEPA tools. The Bio-PEPA plugin which is used for modelling and simulation of Bio-PEPA models is written entirely in Java (based on the Eclipse IDE). Using the same language for the development of our tool means that the two could be easily integrated in the future. 2. Another important reason is that there are several third-party tools 39

54 written for programming in Java which ease the development process. We would like to particularly mention the availability of a Java Application Programming Interface (API) for processing SBML files called libsbml [39]. This has been embedded in our tool to allow us to easily read SBML models. 3. Java is a modern object-oriented programming language. It allows us to build individual parts of our tool (using standalone Java classes) and then bring them together to compose the whole tool. This speeds up the development process considerably. 4. Finally, Java is a highly portable language which can run on a wide variety of platforms. This allows us to easily deploy our tool in different environments. 4.2 Java Classes Design We now explain some of the main classes used in the software tool. Figure 4.1 gives an overview of the main classes of our software. Note that this figure does not show the internal details of the classes. These will be explained along with the corresponding class description below. 1. SBML2BioPEPA: This is the main class of the application. It represents the class which does the actual mapping from a SBML file to a Bio-PEPA file suitable for use in the Bio-PEPA Eclipse plugin. This class does not have any data members. It only accepts from the user the SBML file which is to be converted to Bio-PEPA, and the name of the file in which to write the Bio-PEPA model. Internally, the class uses the SBMLModel and BioPEPAModel classes (described below) for storing the model data. We briefly explain below some of the main methods of this class. 40

55 Figure 4.1: Main classes used in the SBML2BioPEPA software tool CreateAbstractSBMLModel: Uses the libsbml API classes to create an abstract representation of the SBML file in memory. ConvertToBioPEPA: Converts the abstract SBML model returned by the above method into the equivalent abstract Bio- PEPA model in memory. It uses all the helper methods listed below to create the individual parts of the Bio-PEPA abstract model. CreateBioPEPACompartments: Extracts compartment information from the SBML model and stores it in the Bio-PEPA model. CreateBioPEPAParameters: Retrieves parameter information from the SBML model and stores it in the Bio-PEPA model. CreateBioPEPASpecies: Populates species-related information in the Bio-PEPA model after extracting it from the SBML model. CreateBioPEPAFunctionalRates: Converts the kinetic-law of each SBML reaction into the functional rate used in the Bio-PEPA model. 41

56 CreateBioPEPAComponents: Using the information about reactions (such as participating species and their stoichiometries), creates the species sequential Bio-PEPA components as well as the overall model component and stores this in the abstract Bio-PEPA model in memory. 2. SBMLModel: This class represents the abstract model of the SBML file. Internally it uses the libsbml API and its data structures to hold information about the SBML model in memory. 3. LibSBML and LibSBMLInternal classes: These classes are the API wrappers provided by libsbml to manipulate SBML documents. They are used to parse the SBML file and to create a in-memory representation of the corresponding SBML model. 4. BioPEPEEclipsePluginFormatter: This class is used to create a Bio-PEPA file suitable for use in the Bio-PEPA Eclipse plugin. This class queries the BioPEPAModel for information about the model, and then writes this to a Bio-PEPA file. 5. BioPEPAModel: Represents the Bio-PEPA model in an abstract manner in memory, i.e. it contains all those details which can be used to create a well-formed Bio-PEPA model. Internally, this class uses several container classes (listed below) to store all the information about the Bio-PEPA model. 6. BioPEPACompartment: Contains all the information required to represent a compartment in Bio-PEPA, namely it has data members for the name, size and unit of size. 7. BioPEPASpecies: This class represents a Bio-PEPA species. It is able to hold information about the species name, enclosing compartment, initial and maximum concentrations, concentration units, step size and number of levels. 42

57 8. BioPEPAParameter: Represents information about a parameter in Bio-PEPA, i.e. parameter name, value and unit. 9. BioPEPAFunctionalRate: Contains data members to represent information associated with functional rates such as the reaction name and the mathematical formula for the rate of the reaction. 10. BioPEPAComponent: There is one instance of this class associated with each species, which represents the Bio-PEPA sequential component. It contains an array of data members to represent the reactions this species is involved in, and the role and stoichiometry of the species in each reaction. 4.3 Tool Deployment In this section, we describe how the tool we have created is used in practice. We also describe how to deploy it on different operating system platforms. The output of the Java classes we have written is a SBML2BioPEPA.JAR file. This file contains the Java executable code which performs the mapping. In addition, we also have a libsbml.jar file which contains the executable code for the libsbml API. To be able to use our tool, the user should copy both of these files into a single folder. Then the user should issue the following command: java -jar SBML2BioPEPA.jar <InputSBMLFile> <OutputBioPEPAFile> Here, InputSBMLFile represents the SBML file we want to convert to Bio-PEPA. OutputBioPEPAFile represents the filename in which we want the translated model to be stored. 43

58 The output Bio-PEPA file uses the same syntax for representing Bio- PEPA models as the Bio-PEPA Eclipse Plugin; this syntax is described in Appendix C. The tool prints out status information such as which elements were successfully mapped, and if there were any problems during the mapping. 44

59 Chapter 5 Case Studies In this chapter, we discuss the modelling and simulation of two biochemical networks, namely the Synthetic Oscillatory Network model (also called the Repressilator) [40] and the Mitogen Activated Protein Kinase (MAPK) pathway [42]. The SBML models of these biological pathways are obtained from the BioModels model repository [23] and converted to Bio-PEPA using our tool. We begin the chapter by describing each of the biological models in detail. Then, we show how the components in the SBML models are translated into Bio-PEPA using the software tool described in the previous chapter. Finally, the results obtained by simulating the Bio-PEPA models in the Bio-PEPA Eclipse plugin are explained and compared with the results reported in the literature and the results obtained by simulation in COPASI with the same SBML file. 45

60 5.1 Repressilator - A Synthetic Oscillatory Network Description of the model The Repressilator is a synthetic regulatory gene network composed of three genes whose protein products mutually suppress each other. This synthetic network was implemented in E. coli [40]. The model consists of three proteins Lacl from E. coli, TetR from the Figure 5.1: Schematic diagram of the Represillator tetracycline-resistance transposon Tn10 and λ-cl from λ-phage. In order to know the current state of the model, a reporter plasmid was constructed by cloning tet-repressible promoter fused to green fluorescent protein (GFP). The signal from the reporter plasmid enables the behavioral study of Repressilator using fluorescent microscopy. The principle behind this network is that three genes are connected in a feedback loop as shown in Figure 5.1. The protein product of one gene is the repressor for the next gene in the loop. The main events in the working of the Repressilator are: 1. The first protein Lacl inhibits the transcription of the second gene TetR. 2. The protein product of TetR in turn inhibits the expression of the gene λ-cl. 46

61 3. Finally, λ-cl protein inhibits Lacl expression thus completing the cycle. As each gene represses the next gene in the loop, this results in cyclic temporal oscillations in the concentrations of the three proteins in the network. Each of the three proteins participates in the different transcription, translation and degradation reactions in various roles. The authors of [40] suggest that the temporal oscillations are due to the variation in the concentrations of each of its components Mapping Description The SBML model for the Repressilator was downloaded from the BioModels database (BioModels number BIOMD ). Using the tool we implemented, we were successfully able to convert this SBML file into the corresponding Bio-PEPA model. In this section, we describe the components in the SBML file and how they are translated into components of the Bio- PEPA model. The syntax for the Bio-PEPA model is the one used in the Bio-PEPA Eclipse plugin as described in Appendix C. The SBML model file for the Repressilator consists of the following elements: 1. One compartment cell of size species as listed in Table 5.1. The species X, Y and Z are the transcripts of mrna and they express proteins PX, PY and PZ respectively. All the species are located in the compartment cell. The SBML file reports the initial amounts of each species; these values are reported in Table

62 ID Name Compartment InitialAmount X Transcript cell 0 Lacl Y Transcript cell 0 TetR Z Transcript λ- cell 0 cl PX Lactose cell 5 operon repressor PY Tetracycline cell 0 repressor PZ Repressor protein CI cell 15 Table 5.1: List of Species in Repressilator model The corresponding section for the species from the automatically generated Bio-PEPA file is shown below: //Species species PX: M0=5.0,V=cell,H=1,N=2; species PY: M0=0.0,V=cell,H=1,N=2; species PZ: M0=15.0,V=cell,H=1,N=2; species X: M0=0.0,V=cell,H=1,N=2; species Y: M0=0.0,V=cell,H=1,N=2; species Z: M0=0.0,V=cell,H=1,N=2; The Bio-PEPA file shows the list of species with species name, enclosing compartment name, and step size H = 1 and number of levels N = global parameters used throughout the model; the parameter and their values are listed in Table 5.2. The corresponding list of parameters mapped to Bio-PEPA are: 48

63 Parameter name Value Ratio of protein to mrna decay rates 0.2 Leakiness in protein copies per promoter 0.25 Protein copies per promoter 250 Translation efficiency, proteins per transcript 20 Hill Constant 2 Number of Repressors, half maximal repression 40 tau mrna 2 Table 5.2: List of Parameters in Repressilator model //Parameters beta = 0.2; alpha0 = 0.25; alpha = 250.0; eff = 20.0; n = 2; KM = 40.0; tau_mrna = 2.0; irreversible reactions and their kinetic-laws. These reactions are the transcription of each gene in the network in mrna, the translation of mrna into the corresponding protein and the degradation of both mrna molecules and proteins. The list of reactions, their participating species, role (reactant, product or modifier) and kinetic-laws are listed in Table 5.3. There are complex kinetic-laws describing the transcription with inhibition and some reactions are described by mass-action kinetics. 49

64 Id Name Reactant Product Modifier Kinetic-law 1 Degradation X - - X / tau mrna of LacI transcripts 2 Degradation Y - - Y / tau mrna of TetR transcripts 3 Degradation Z - - Z / tau mrna of CI transcripts 4 Translation of - PX X X * KM * beta / (tau mrna * eff LacI 5 Translation of - PY Y Y * KM * beta/(tau mrna * eff) TetR 6 Translation of CI - PZ Z Z * KM * beta / (tau mrna * eff) 7 Degradation PX - - PX * beta / tau mrna of LacI 8 Degradation PY - - PY * beta / tau mrna of tetr 9 Degradation PZ - - PZ * beta / tau mrna of CI 10 Transcription of LacI - X PZ alpha0 * eff / tau mrna + alpha * eff * (KM * KM) / (tau mrna 11 Transcription of tetr 12 Transcription of CI * ((KM * KM) + (PZ * PZ) - Y PX alpha0 * eff / tau mrna + alpha * eff * (KM * KM) / (tau mrna * ((KM * KM) + (PX * PX) - Z PY alpha0 * eff / tau mrna + alpha * eff * (KM * KM) / (tau mrna * ((KM * KM) + (PY * PY) Table 5.3: List of Reactions in Repressilator model 50

65 The kinetic-law of each reaction is mapped to the corresponding functional rate in Bio-PEPA. //Functional Rates kineticlawof Alpha_0 : X / tau_mrna; kineticlawof Alpha_1 : Y / tau_mrna; kineticlawof Alpha_2 : Z / tau_mrna; kineticlawof Alpha_3 : X * KM * beta / (tau_mrna * eff); kineticlawof Alpha_4 : Y * KM * beta / (tau_mrna * eff); kineticlawof Alpha_5 : Z * KM * beta / (tau_mrna * eff); kineticlawof Alpha_6 : PX * beta / tau_mrna; kineticlawof Alpha_7 : PY * beta / tau_mrna; kineticlawof Alpha_8 : PZ * beta / tau_mrna; kineticlawof Alpha_9 : alpha0 * eff / tau_mrna + alpha * eff * (KM * KM) / (tau_mrna * ((KM * KM) + (PZ * PZ))); kineticlawof Alpha_10 : alpha0 * eff / tau_mrna + alpha * eff * (KM * KM) / (tau_mrna * ((KM * KM) + (PX * PX))); kineticlawof Alpha_11 : alpha0 * eff / tau_mrna + alpha * eff * (KM * KM) / (tau_mrna * ((KM * KM) + (PY * PY))); The information about reactions in the SBML file is used to create the sequential components and the model component in Bio-PEPA as described in Chapter 3. The species definitions and the complete model component in the obtained Bio-PEPA file are listed below: //Components PX = (Alpha_3,1.0) >> PX + (Alpha_6,1.0) << PX + (Alpha_10,1.0) (.) PX; X = (Alpha_0,1.0) << X + (Alpha_3,1.0) (.) X 51

66 + (Alpha_9,1.0) >> X; Y = (Alpha_1,1.0) << Y + (Alpha_4,1.0) (.) Y + (Alpha_10,1.0) >> Y; PY = (Alpha_4,1.0) >> PY + (Alpha_7,1.0) << PY + (Alpha_11,1.0) (.) PY; PZ = (Alpha_5,1.0) >> PZ + (Alpha_8,1.0) << PZ + (Alpha_9,1.0) (.) PZ; Z = (Alpha_2,1.0) << Z + (Alpha_5,1.0) (.) Z + (Alpha_11,1.0) >> Z; //System Model (PX[1] <Alpha_10, Alpha_3> (Y[1] <Alpha_10, Alpha_4> (PY[1] <Alpha_11, Alpha_4> (X[1] <Alpha_3, Alpha_9> (PZ[1] <Alpha_5, Alpha_9> (Z[1])))))) We see that the reactions give us the information about role of the species which are incorporated in the species definitions. For example, we see that protein PX acts as a reactant in reaction 6, as a product in reaction 4 and modifier in reaction 11. We also see that information about stoichiometry of a species in each reaction has been included in the Bio-PEPA model. The complete Bio-PEPA model file for the Repressilator is listed in Appendix A. 52

67 5.1.3 Simulation of Bio-PEPA Repressilator model We used the Bio-PEPA Eclipse plugin to simulate the Bio-PEPA model of the Repressilator. The plugin is capable of performing ODE and stochastic simulations of Bio-PEPA models. We had to make minor changes to the generated Bio-PEPA Repressilator model file before it could be used in the plugin. These changes were needed because the kinetic-law in the SBML model uses the pow function which is not currently supported by the plugin. We replaced the pow function with the equivalent numerical value. We then simulated the model using the Gibson-Bruck Algorithm for stochastic simulation [41]. The simulation was run for 1000 time steps with the number of independent replications set to 10. The results obtained from this simulation are shown in Figure 5.2. Figure 5.2: Stochastic simulation results of oscillations in Repressilator network using Bio-PEPA plugin The original SBML model file that we downloaded from the BioModels repository was also imported into COPASI [29] for simulation, so we could compare 53

68 the results we obtained from the two tools. In COPASI, we again used the Gibson-Bruck stochastic simulation algorithm, with the same parameter settings as that of the Bio-PEPA Plugin. The results obtained are shown in Figure 5.3. Figure 5.3: Stochastic simulation results of oscillations in Repressilator network using COPASI tool The results show the oscillations in the levels of three proteins, lactose operon repressor, tetracycline repressor and repressor protein Cl. The results from Bio-PEPA plugin were found to be in good agreement both with the results obtained in COPASI and the results reported in [40]. This gives us confidence in the correctness of our generated model. We can say that our tool was correctly able to generate the Bio-PEPA model from the SBML file and that our mapping is correct. 54

69 5.2 Mitogen Activated Proetin Kinases Model Description of the model Signal transduction pathways enable cells to respond to external stimuli in an appropriate manner. The Mitogen Activated Protein Kinase (MAPK) signalling pathways play an important role in signal transduction and are a key component in a variety of cellular functions such as gene expression, mitosis, differentiation and cell apoptosis [43]. They are evolutionarily conserved in all eukaryotes. MAPK cascades are generally considered to be of three levels, where the kinase at each level phosphorylates the kinase at the next level. MAP-kinases are proline-directed serine/threonine kinases that are activated by dual phosphorylation in response to diverse extracellular stimuli. The dual phosphorylation occurs in the activation domain of MAPKs on the threonine and tyrosine residues. The dephosphorylation leads to inactivation of the kinase. The MAPK cascade is shown in Figure 5.4 and the basic steps involved in the cascade are reported below. 1. The interaction between small proteins and MAP Kinase Kinase Kinase (MAPKKK) leads to the activation of kinase cascade and MAPKKK is phosphorylated. 2. The phosphorylated MAPKKK in turn phosphorylates MAP Kinase Kinase (MAPKK). 3. MAPKK then phosphorylates MAPK in the terminal level of the cascade. The authors of [42] claim that amplification of signal in MAPK cascades is one of the physiological functions of kinases. Small number of signalling molecules cause the conversion of large number of target molecules in these systems. They analysed the properties that lead to sustained oscilations in the biological systems by means of kinetic models of MAPK cascade. 55

70 Figure 5.4: The Mitogen Activated Protein Kinase cascade Mapping Description The SBML model file for the MAPK cascade was downloaded from the BioModels database [23] (Biomodel number BIOMD ). The model was sucessfully converted to Bio-PEPA model using our tool, SBML2BioPEPA. Below, we give a description of the elements of the SBML model and the corresponding elements of the Bio-PEPA model that was obtained. The syntax for the Bio-PEPA model is the one used in the Bio-PEPA Eclipse plugin as described in Appendix C. The SBML model file for the MAPK pathway consists of the following elements: 1. 1 compartment uvol of size species as listed in Table 5.4. The initial concentrations of the species 56

71 ID Compartment Initial Concentration MKKK uvol 90.0 MKKK P uvol 10.0 MKK uvol MKK P uvol 10.0 MKK PP uvol 10.0 MAPK uvol MAPK P uvol 10.0 MAPK PP uvol 10.0 Table 5.4: List of Species in MAPK model are also given. The corresponding definition of species in the Bio-PEPA file is as follows: //Species species MKKK: M0=90.0,V=uVol,H=1,N=2; species MKKK_P: M0=10.0,V=uVol,H=1,N=2; species MKK: M0=280.0,V=uVol,H=1,N=2; species MKK_P: M0=10.0,V=uVol,H=1,N=2; species MKK_PP: M0=10.0,V=uVol,H=1,N=2; species MAPK: M0=280.0,V=uVol,H=1,N=2; species MAPK_P: M0=10.0,V=uVol,H=1,N=2; species MAPK_PP: M0=10.0,V=uVol,H=1,N=2; parameters as listed in Table 5.5. In this model, most of the parameters are local to the particular reactions. These parameters are used in evaluating the kinetic-law expressions of those reactions. The corresponding list of parameter in Bio-PEPA are: //Parameters 57

72 Parameter name Value uvol param 1.0 V1 2.5 Ki 9.0 n 1.0 K V KK2 8.0 k KK k KK V KK V KK k KK k KK V9 0.5 KK V KK Table 5.5: List of Parameters in MAPK model 58

73 uvol_param = 1.0; V1 = 2.5; Ki = 9.0; K1 = 10.0; V2 = 0.25; KK2 = 8.0; k3 = 0.025; KK3 = 15.0; k4 = 0.025; KK4 = 15.0; V5 = 0.75; KK5 = 15.0; V6 = 0.75; KK6 = 15.0; k7 = 0.025; KK7 = 15.0; k8 = 0.025; KK8 = 15.0; V9 = 0.5; KK9 = 15.0; V10 = 0.5; KK10 = 15.0; Reactions as listed in Table 5.6. The table also lists the reactants, products and modifiers along with the kinetic-laws of the reactions. The kinases with dual specificity follow the Michaelis-Menten ratelaw [42]. Therefore, the rate equations in the cascade are shown by Michaelis-Menten kinetics. The catalytic rate constants and maximal enzymatic rates for each reaction are given in the list of parameters. The functional rates corresponding to the kinetic-laws are expressed in the Bio-PEPA file as shown: 59

74 Reaction name Reactant Product Modifier Kinetic-law MAPKKK activation MKKK / ((1 + MKKK MKKK P MAPK PP uvol * V1 * pow(mapk PP / Ki, n)) * (K1 + MAPKKK inactivation phosphorylation of MAPKK phosphorylation of MAPKK-P dephosphorylation of MAPKK-PP MKKK)) MKKK P MKKK - uvol * V2 * MKKK P / (KK2 + MKKK P) MKK MKK P MKKK P uvol * k3 * MKKK P * MKK / (KK3 + MKK) MKK P MKK PP MKKK P uvol * k4 * MKKK P * MKK P / (KK4 + MKK P) MKK PP MKK P - uvol * V5 * MKK PP / (KK5 + MKK PP) dephosphorylation MKK P MKK - uvol * V6 * MKK P of MAPKK-P / (KK6 + MKK P) phosphorylation of MAPK MAPK MAPK P MKK PP uvol * k7 * MKK PP * MAPK / (KK7 + phosphorylation of MAPK-P dephosphorylation of MAPK-PP dephosphorylation of MAPK-P MAPK) MAPK P MAPK PP MKK PP uvol * k8 * MKK PP * MAPK P / (KK8 + MAPK P) MAPK PP MAPK P - uvol * V9 * MAPK PP / (KK9 + MAPK PP) MAPK P MAPK - uvol * V10 * MAPK P / (KK10 + MAPK P) Table 5.6: List of Reactions in MAPK model 60

75 //Functional Rates kineticlawof Alpha_0 : uvol_param * V1 * MKKK / ( (1 + MAPK_PP / Ki) * (K1 + MKKK)); kineticlawof Alpha_1 : uvol_param * V2 * MKKK_P / (KK2 + MKKK_P); kineticlawof Alpha_2 : uvol_param * k3 * MKKK_P * MKK / (KK3 + MKK); kineticlawof Alpha_3 : uvol_param * k4 * MKKK_P * MKK_P / (KK4 + MKK_P); kineticlawof Alpha_4 : uvol_param * V5 * MKK_PP / (KK5 + MKK_PP); kineticlawof Alpha_5 : uvol_param * V6 * MKK_P / (KK6 + MKK_P); kineticlawof Alpha_6 : uvol_param * k7 * MKK_PP * MAPK / (KK7 + MAPK); kineticlawof Alpha_7 : uvol_param * k8 * MKK_PP * MAPK_P / (KK8 + MAPK_P); kineticlawof Alpha_8 : uvol_param * V9 * MAPK_PP / (KK9 + MAPK_PP); kineticlawof Alpha_9 : uvol_param * V10 * MAPK_P / (KK10 + MAPK_P); We construct the species and model components from the information about the reactions. We use the suffix P to indicate that the species is phosphorylated once and, similarly, the suffix P P to indicate that the species is phosphorylated twice. We can see the different roles played by the same species in different reactions along with the stoichiometric information. The synchronization of species over action types is given by the model component. The species and model component of the MAPK cascade in the Bio-PEPA file is as reported below: 61

76 //Components MAPK = (Alpha_6,1.0) << MAPK + (Alpha_9,1.0) >> MAPK; MKK_P = (Alpha_2,1.0) >> MKK_P + (Alpha_3,1.0) << MKK_P + (Alpha_4,1.0) >> MKK_P + (Alpha_5,1.0) << MKK_P; MKK_PP = (Alpha_3,1.0) >> MKK_PP + (Alpha_4,1.0) << MKK_PP + (Alpha_6,1.0) (.) MKK_PP + (Alpha_7,1.0) (.) MKK_PP; MKKK_P = (Alpha_0,1.0) >> MKKK_P + (Alpha_1,1.0) << MKKK_P + (Alpha_2,1.0) (.) MKKK_P + (Alpha_3,1.0) (.) MKKK_P; MKK = (Alpha_2,1.0) << MKK + (Alpha_5,1.0) >> MKK; MAPK_PP = (Alpha_0,1.0) (.) MAPK_PP + (Alpha_7,1.0) >> MAPK_PP + (Alpha_8,1.0) << MAPK_PP; MAPK_P = (Alpha_6,1.0) >> MAPK_P + (Alpha_7,1.0) << MAPK_P + (Alpha_8,1.0) >> MAPK_P + (Alpha_9,1.0) << MAPK_P; MKKK = (Alpha_0,1.0) << MKKK + (Alpha_1,1.0) >> MKKK; //System Model (MAPK[1] <Alpha_6, Alpha_9> (MKK_P[1] <Alpha_2, Alpha_3, Alpha_4, Alpha_5> (MKK_PP[1] <Alpha_3, Alpha_4, Alpha_6, Alpha_7> 62

77 (MKKK_P[1] <Alpha_0, Alpha_1, Alpha_2, Alpha_3> (MKK[1] <Alpha_2, Alpha_5> (MAPK_PP[1] <Alpha_0, Alpha_7, Alpha_8> (MAPK_P[1] <Alpha_6, Alpha_7, Alpha_8, Alpha_9> (MKKK[1])))))))) Simulation of Bio-PEPA MAPK model We simulated the generated Bio-PEPA model of MAPK using the Bio-PEPA Eclipse Plugin. The simulation was carried out with the Gibson-Bruck stochastic simulation algorithm [41]. The start and stop time were set to 0 and 5000 respectively, and the number of replications was set to 10. The results of the simulation are shown in Figure 5.5. The SBML model file was also imported into COPASI and we again ran the Gibson-Bruck algorithm using the same parameters values as in the Bio-PEPA plugin simulation. The simulation results using COPASI are shown in Figure 5.6 Figure 5.5: Stochastic simulation results of MAPK using Bio-PEPA Plugin The dynamics of the MAPK, phosphorylated MAPK, biphosphorylated MAPK and MAPKKKK can be observed from the figures. The cascade can occur 63

78 Figure 5.6: Stochastic simulation results of MAPK using COPASI tool in low and high phosphorylation levels. We see that the concentration of MAPK decreases as the concentration of MAPK-PP increases and a pattern of oscillation is observed. The simulation results obtained from COPASI and from the Bio-PEPA plugin agree to a large extent. The observed behaviour of the species in both the simulations are in agreement with each other. This affirms the correctness of our mapping and our tool in generating Bio-PEPA process algebra models. 64